Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

TL;DR

This paper classifies all possible gradings by abelian groups on certain classical simple Lie and associative superalgebras over algebraically closed fields, including fine gradings and those induced from associative superalgebras.

Contribution

It provides a comprehensive classification of gradings on superalgebras M(m,n), A(m,n), and P(n), including fine gradings and gradings induced from associative structures.

Findings

Classified fine gradings up to equivalence.

Classified G-gradings up to isomorphism.

Connected gradings on classical Lie superalgebras.

Abstract

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n \geq 2$, and on the simple associative superalgebras $M(m,n)$, $m, n \geq 1$, over an algebraically closed field: fine gradings up to equivalence and $G$-gradings, for a fixed group $G$, up to isomorphism. As a corollary, we also classify up to isomorphism the $G$-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from $G$-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be $0$.